How do you solve..?

Question

A rectangular lawn measures 7m by 5m. A uniform border of flowers is to be palnted along two adjacent sides of hte lawn. If the flowers that have been purchased will cover an area of 6.25m(squared), how wide is the border?

The answer is 0.5m

Answer 1

Let us set 'W' as the width of the flower border.

5m
,-----^-----,
Wm
+-+-------+............\
|...|........|............. \ 7m
+-+-------+..\ Wm.../
+-+-------+../......../

We know that the maximum area of flowers will cover 6.25 m².
How do you calculate this area of flowers for your lawn:
- On the one side of your lawn it will be 7m by W m and
- On the other side of your lawn it will be (5-W) m by W m.
... the addition of which two numbers will give you the total area of flowers.

So you write:
6.25 = 7xW + (5-W)xW
6.25 = 7xW + 5xW - W²
6.25 = (7+5)xW - W²
6.25 - 12xW + W² = 0

This gives you a quadratic equation, which you can solve or factorize this way:
- you have first to remember that (W-A)² = W² - 2xWxA + A²
- you imagine then that W² -12xW is the start of a term of the form W² - 2xWxA
- which is in this case W² -2xWx6
- so you add and subtract 6² to your equation:
6.25 -6² + 6² -12xW + W² = 0
(6.25 -36) + (W-6)² = 0
(W-6)² - (29.75) = 0
- to factorize this further you have now to remember that B²-C² = (B-C)x(B+C)
- since 29.75 is positive, you can say that (29.75) = ( √(29.75) )²
(W-6)² - ( √(29.75) )² = 0
( (W-6) - √(29.75) )x( (W-6) + √(29.75) ) = 0
( W - 6 - √29.75 )x( W - 6 + √29.75 ) = 0
( W - (6+√29.75) )x( W - (6-√29.75) ) = 0

This product is zero if one or both terms are zero, which leads to two possibles solutions to you quadratic equation:
W = (6+√29.75) ~ 6 + 5.4543 = 11.4543
W = (6-√29.75) ~ 6 - 5.4543 = 0.5457

It looks like both would be correct as these W solutions (each representing a width for your flower band) are possible. .. let us verify this with the initial problem, by calculating the area of flower in your lawn:
- with the first solution, W ~ 11.4543
7x(11.4543) + (5-11.4543)x11.4543 ~ 6.25
- with the second solution, W ~ 0.5457
7x(0.5457) + (5-0.5457)x0.5457 ~ 6.25

What is wrong then ?
You have to note that when we wrote (5-W)xW, we implicitly supposed that W was smaller than 5 m !

So your valid solution is 0.5457 (give or take, depending of your calculator).

Answer 2

Let x be the width of the border. The one side will be 7m by x m and the other side will be 5-x m by x m. So one side will have area 7x m² and the other side will have area
5x-x² m² for a total of 12x - x² m² which is 6.25 m² so
6.25 = 12x - x² or x²-12x+6.25 = 0
Solve for x and your done. (If you don't know how to solve a standard form quadratic equation using the quadratic formula, it's high time you learned ☺)


Doug

Answer 3

The border width is x
One strip along one side, measures 5' by x.
The other strip measures (7-x)' by x

6.25 m² = (5x) + (7-x)(x)
6.25 = 5x + 7x - x²
x² - 12x + 6.25 = 0